Last updated: 2022-01-20

Checks: 4 2

Knit directory: Padgett-Dissertation/

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# Load packages & utility functions
source("code/load_packages.R")
source("code/load_utility_functions.R")
# environment options
options(scipen = 999, digits=3)

Describing the Observed Data

# Load diffIRT package with data
library(diffIRT)
data("extraversion")
mydata <- na.omit(extraversion)

# separate data then recombine
d1 <- mydata %>%
  as.data.frame() %>%
  select(contains("X"))%>%
  mutate(id = 1:n()) %>%
  pivot_longer(
    cols=contains("X"),
    names_to = c("item"),
    values_to = "Response"
  ) %>%
  mutate(
    item = ifelse(nchar(item)==4,substr(item, 3,3),substr(item, 3,4))
  )
d2 <- mydata %>%
  as.data.frame() %>%
  select(contains("T"))%>%
  mutate(id = 1:n()) %>%
  pivot_longer(
    cols=starts_with("T"),
    names_to = c("item"),
    values_to = "Time"
  ) %>%
  mutate(
    item = ifelse(nchar(item)==4,substr(item, 3,3),substr(item, 3,4))
  )
dat <- left_join(d1, d2)

Joining, by = c("id", "item")

dat_sum <- dat %>%
  select(item, Response, Time) %>%
  group_by(item) %>%
  summarize(
    M1 = mean(Response, na.rm=T),
    Mt = mean(Time, na.rm=T),
    SDt = sd(Time, na.rm=T),
    Mlogt = mean(log(Time), na.rm=T),
  )

colnames(dat_sum) <-
  c(
    "Item",
    "Proportion Endorsed",
    "Mean Response Time",
    "SD Response Time",
    "Mean Log Response Time"
  )

kable(dat_sum, format = "html", digits = 3) %>%
  kable_styling(full_width = T)

Item	Proportion Endorsed	Mean Response Time	SD Response Time	Mean Log Response Time
1	0.739	1.488	0.805	0.288
10	0.866	0.979	0.520	-0.115
2	0.535	1.354	0.648	0.208
3	0.852	1.115	0.632	0.002
4	0.923	1.001	0.664	-0.114
5	0.542	1.301	0.706	0.163
6	0.901	1.255	0.682	0.119
7	0.944	1.143	0.546	0.054
8	0.965	1.067	0.575	-0.030
9	0.824	1.728	0.745	0.463

# covariance among items
kable(cov(mydata[,colnames(mydata) %like% "X"]), digits = 3) %>%
  kable_styling(full_width = T)

	X[1]	X[2]	X[3]	X[4]	X[5]	X[6]	X[7]	X[8]	X[9]	X[10]
X[1]	0.194	-0.001	0.039	0.029	0.000	0.002	0.014	0.005	0.011	0.015
X[2]	-0.001	0.251	0.023	0.006	0.077	0.011	0.002	0.012	0.031	0.030
X[3]	0.039	0.023	0.127	0.038	0.024	0.028	0.020	0.016	0.016	0.051
X[4]	0.029	0.006	0.038	0.072	0.014	0.006	0.017	0.019	0.029	0.025
X[5]	0.000	0.077	0.024	0.014	0.250	0.004	0.017	0.005	0.032	0.031
X[6]	0.002	0.011	0.028	0.006	0.004	0.090	0.009	0.011	0.004	0.015
X[7]	0.014	0.002	0.020	0.017	0.017	0.009	0.054	0.019	0.004	0.007
X[8]	0.005	0.012	0.016	0.019	0.005	0.011	0.019	0.034	0.008	0.009
X[9]	0.011	0.031	0.016	0.029	0.032	0.004	0.004	0.008	0.146	0.033
X[10]	0.015	0.030	0.051	0.025	0.031	0.015	0.007	0.009	0.033	0.117

# correlation matrix
psych::polychoric(mydata[,colnames(mydata) %like% "X"])

Warning in cor.smooth(mat): Matrix was not positive definite, smoothing was done

Call: psych::polychoric(x = mydata[, colnames(mydata) %like% "X"])
Polychoric correlations 
      X[1]  X[2]  X[3]  X[4]  X[5]  X[6]  X[7]  X[8]  X[9]  X[10]
X[1]   1.00                                                      
X[2]  -0.01  1.00                                                
X[3]   0.45  0.24  1.00                                          
X[4]   0.50  0.11  0.70  1.00                                    
X[5]   0.00  0.46  0.26  0.23  1.00                              
X[6]   0.04  0.15  0.50  0.21  0.06  1.00                        
X[7]   0.32  0.05  0.52  0.58  0.36  0.32  1.00                  
X[8]   0.18  0.38  0.57  0.71  0.17  0.48  0.78  1.00            
X[9]   0.12  0.29  0.24  0.55  0.31  0.08  0.13  0.31  1.00      
X[10]  0.19  0.34  0.69  0.54  0.35  0.32  0.22  0.39  0.47  1.00

 with tau of 
           1
X[1]  -0.642
X[2]  -0.088
X[3]  -1.046
X[4]  -1.422
X[5]  -0.106
X[6]  -1.290
X[7]  -1.586
X[8]  -1.809
X[9]  -0.930
X[10] -1.109

Model 1: Traditional IFA

Model details

cat(read_file(paste0(w.d, "/code/study_4/model_1.txt")))

model {
### Model
  for(p in 1:N){
    for(i in 1:nit){
      # data model
      y[p,i] ~ dbern(pi[p,i,2])

      # LRV
      ystar[p,i] ~ dnorm(lambda[i]*eta[p], 1)

      # Pr(nu = 2)
      pi[p,i,2] = phi(ystar[p,i] - tau[i,1])
      # Pr(nu = 1)
      pi[p,i,1] = 1 - phi(ystar[p,i] - tau[i,1])

    }
  }
  ### Priors
  # person parameters
  for(p in 1:N){
    eta[p] ~ dnorm(0, 1) # latent ability
  }

  for(i in 1:nit){
    # Thresholds
    tau[i, 1] ~ dnorm(0.0,0.1)
    # loadings
    lambda[i] ~ dnorm(0, .44)T(0,)
    # LRV total variance
    # total variance = residual variance + fact. Var.
    theta[i] = 1 + pow(lambda[i],2)
    # standardized loading
    lambda.std[i] = lambda[i]/pow(theta[i],0.5)
  }

  # compute omega
  lambda_sum[1] = lambda[1]
  for(i in 2:nit){
    #lambda_sum (sum factor loadings)
    lambda_sum[i] = lambda_sum[i-1]+lambda[i]
  }
  reli.omega = (pow(lambda_sum[nit],2))/(pow(lambda_sum[nit],2)+nit)
}

Model results

# Save parameters
jags.params <- c("tau", "lambda", "theta", "reli.omega", "lambda.std")
# initial-values
jags.inits <- function(){
    list(
      "tau"=matrix(c(-0.64, -0.09, -1.05, -1.42, -0.11, -1.29, -1.59, -1.81, -0.93, -1.11),
                   ncol=1, nrow=10),
      "lambda"=rep(0.7,10),
      "eta"=rnorm(142),
      "ystar"=matrix(c(0.7*rep(rnorm(142),10)), ncol=10)
    )
  }
# data
jags.data <- list(y = mydata[,1:10],
               N = nrow(mydata),
               nit = 10)
model.fit <-  R2jags::jags(
  model = paste0(w.d, "/code/study_4/model_1.txt"),
  parameters.to.save = jags.params,
  inits = jags.inits,
  data = jags.data,
  n.chains = 4,
  n.burnin = 5000,
  n.iter = 10000
)

module glm loaded

Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 1420
   Unobserved stochastic nodes: 1582
   Total graph size: 8742

Initializing model

print(model.fit, width=1000)

Inference for Bugs model at "C:/Users/noahp/Documents/GitHub/Padgett-Dissertation/code/study_4/model_1.txt", fit using jags,
 4 chains, each with 10000 iterations (first 5000 discarded), n.thin = 5
 n.sims = 4000 iterations saved
               mu.vect sd.vect    2.5%     25%     50%     75%   97.5% Rhat n.eff
lambda[1]        0.508   0.233   0.099   0.341   0.496   0.658   1.003 1.01   430
lambda[2]        0.657   0.251   0.215   0.480   0.643   0.813   1.184 1.00   650
lambda[3]        1.996   0.660   0.919   1.524   1.929   2.372   3.516 1.03   130
lambda[4]        1.657   0.595   0.690   1.225   1.589   2.033   3.010 1.02   160
lambda[5]        0.647   0.254   0.180   0.471   0.635   0.806   1.181 1.01   480
lambda[6]        0.828   0.346   0.228   0.586   0.804   1.034   1.598 1.01   350
lambda[7]        1.047   0.432   0.268   0.753   1.015   1.306   2.027 1.01   450
lambda[8]        1.533   0.612   0.564   1.104   1.449   1.864   3.019 1.00   760
lambda[9]        0.749   0.305   0.234   0.540   0.722   0.925   1.452 1.00   580
lambda[10]       1.567   0.513   0.742   1.205   1.504   1.848   2.779 1.01   310
lambda.std[1]    0.431   0.159   0.099   0.323   0.444   0.550   0.708 1.01   430
lambda.std[2]    0.525   0.145   0.210   0.432   0.541   0.631   0.764 1.00   700
lambda.std[3]    0.869   0.075   0.677   0.836   0.888   0.921   0.962 1.02   330
lambda.std[4]    0.823   0.099   0.568   0.775   0.846   0.897   0.949 1.01   210
lambda.std[5]    0.519   0.149   0.177   0.426   0.536   0.628   0.763 1.01   590
lambda.std[6]    0.601   0.160   0.223   0.506   0.627   0.719   0.848 1.01   400
lambda.std[7]    0.679   0.159   0.259   0.602   0.712   0.794   0.897 1.01   630
lambda.std[8]    0.797   0.119   0.491   0.741   0.823   0.881   0.949 1.02   540
lambda.std[9]    0.569   0.153   0.228   0.475   0.586   0.679   0.824 1.00   710
lambda.std[10]   0.816   0.089   0.596   0.769   0.833   0.879   0.941 1.01   350
reli.omega       0.922   0.022   0.875   0.911   0.926   0.937   0.953 1.01   370
tau[1,1]        -0.955   0.180  -1.331  -1.073  -0.950  -0.830  -0.622 1.00  1400
tau[2,1]        -0.132   0.167  -0.462  -0.242  -0.131  -0.019   0.198 1.00   910
tau[3,1]        -2.499   0.580  -3.905  -2.823  -2.425  -2.082  -1.613 1.03   140
tau[4,1]        -3.059   0.645  -4.570  -3.397  -2.975  -2.606  -2.052 1.02   200
tau[5,1]        -0.151   0.166  -0.481  -0.261  -0.150  -0.039   0.169 1.00  4000
tau[6,1]        -2.126   0.304  -2.803  -2.311  -2.100  -1.912  -1.616 1.01   510
tau[7,1]        -2.802   0.456  -3.846  -3.063  -2.748  -2.479  -2.053 1.00  1600
tau[8,1]        -3.767   0.812  -5.775  -4.192  -3.617  -3.195  -2.551 1.00  2200
tau[9,1]        -1.485   0.230  -1.971  -1.621  -1.474  -1.328  -1.080 1.01   470
tau[10,1]       -2.324   0.460  -3.404  -2.584  -2.260  -2.000  -1.601 1.01   610
theta[1]         1.313   0.275   1.010   1.116   1.246   1.433   2.006 1.00   620
theta[2]         1.494   0.367   1.046   1.230   1.413   1.662   2.401 1.00   650
theta[3]         5.421   2.964   1.845   3.322   4.721   6.626  13.360 1.03   100
theta[4]         4.098   2.268   1.476   2.501   3.524   5.133  10.063 1.02   160
theta[5]         1.483   0.367   1.032   1.222   1.403   1.650   2.395 1.01   340
theta[6]         1.806   0.679   1.052   1.344   1.646   2.069   3.552 1.01   360
theta[7]         2.283   1.040   1.072   1.567   2.030   2.705   5.110 1.01   340
theta[8]         3.724   2.237   1.318   2.220   3.100   4.475  10.116 1.00  1200
theta[9]         1.655   0.540   1.055   1.291   1.522   1.855   3.108 1.01   450
theta[10]        3.720   1.872   1.550   2.451   3.263   4.414   8.724 1.01   300
deviance       704.721  30.592 647.942 684.707 703.635 723.982 763.976 1.00  1600

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 467.4 and DIC = 1172.1
DIC is an estimate of expected predictive error (lower deviance is better).

Posterior Distribution Summary

# extract for plotting
jags.mcmc <- as.mcmc(model.fit)
a <- colnames(as.data.frame(jags.mcmc[[1]]))
fit.mcmc <- data.frame(as.matrix(jags.mcmc, chains = T, iters = T))
colnames(fit.mcmc) <- c("chain", "iter", a)
fit.mcmc.ggs <- ggmcmc::ggs(jags.mcmc) # for GRB plot

Categroy Thresholds (\(\tau\))

# tau
bayesplot::mcmc_areas(fit.mcmc, regex_pars = "tau", prob = 0.8); ggsave("fig/study4_model1_tau_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "tau"); ggsave("fig/study4_model1_tau_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "tau"); ggsave("fig/study4_model1_tau_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "tau"); ggsave("fig/study4_model1_tau_grb.pdf")

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Factor Loadings (\(\lambda\))

bayesplot::mcmc_areas(fit.mcmc, regex_pars = "lambda", prob = 0.8)

bayesplot::mcmc_acf(fit.mcmc, regex_pars = "lambda")

bayesplot::mcmc_trace(fit.mcmc, regex_pars = "lambda")

ggmcmc::ggs_grb(fit.mcmc.ggs, family = "lambda")

bayesplot::mcmc_areas(fit.mcmc, regex_pars = "lambda.std", prob = 0.8); ggsave("fig/study4_model1_lambda_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "lambda.std"); ggsave("fig/study4_model1_lambda_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "lambda.std"); ggsave("fig/study4_model1_lambda_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "lambda.std"); ggsave("fig/study4_model1_lambda_grb.pdf")

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Latent Response Total Variance (\(\theta\))

bayesplot::mcmc_areas(fit.mcmc, regex_pars = "theta", prob = 0.8); ggsave("fig/study4_model1_theta_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "theta"); ggsave("fig/study4_model1_theta_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "theta"); ggsave("fig/study4_model1_theta_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "theta"); ggsave("fig/study4_model1_theta_grb.pdf")

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Factor Reliability Omega (\(\omega\))

bayesplot::mcmc_areas(fit.mcmc, regex_pars = "reli.omega", prob = 0.8); ggsave("fig/study4_model1_omega_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "reli.omega"); ggsave("fig/study4_model1_omega_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "reli.omega"); ggsave("fig/study4_model1_omega_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "reli.omega"); ggsave("fig/study4_model1_omega_grb.pdf")

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# extract omega posterior for results comparison
extracted_omega <- data.frame(model_1 = fit.mcmc$reli.omega)
write.csv(x=extracted_omega, file=paste0(getwd(),"/data/study_4/extracted_omega_m1.csv"))

Posterior Predictive Distributions

# Posterior Predictive Check
Niter <- 200
model.fit$model$recompile()

Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 1420
   Unobserved stochastic nodes: 1582
   Total graph size: 8742

Initializing model

fit.extra <- rjags::jags.samples(model.fit$model, variable.names = "pi", n.iter = Niter)
N <- model.fit$model$data()[["N"]]
nit <- 10
nchain=4
C <- 2
n <- i <- iter <- ppc.row.i <- 1
y.prob.ppc <- array(dim=c(Niter*nchain, nit, C))
for(chain in 1:nchain){
  for(iter in 1:Niter){
    # initialize simulated y for this iteration
    y <- matrix(nrow=N, ncol=nit)
    # loop over item
    for(i in 1:nit){
      # simulated data for item i for each person
      for(n in 1:N){
        y[n,i] <- sample(1:C, 1, prob = fit.extra$pi[n, i, 1:C, iter, chain])
      }
      # computer proportion of each response category
      for(c in 1:C){
        y.prob.ppc[ppc.row.i,i,c] <- sum(y[,i]==c)/N
      }
    }
    
    # update row of output
    ppc.row.i = ppc.row.i + 1
  }
}

yppcmat <- matrix(c(y.prob.ppc), ncol=1)
z <- expand.grid(1:(Niter*nchain), 1:nit, 1:C)
yppcmat <- data.frame(  iter = z[,1], nit=z[,2], C=z[,3], yppc = yppcmat)

ymat <- model.fit$model$data()[["y"]]
y.prob <- matrix(ncol=C, nrow=nit)
for(i in 1:nit){
  for(c in 1:C){
    y.prob[i,c] <- sum(ymat[,i]==c-1)/N
  }
}
yobsmat <- matrix(c(y.prob), ncol=1)
z <- expand.grid(1:nit, 1:C)
yobsmat <- data.frame(nit=z[,1], C=z[,2], yobs = yobsmat)
plot.ppc <- full_join(yppcmat, yobsmat)

Joining, by = c("nit", "C")

p <- plot.ppc %>%
  mutate(C    = as.factor(C),
         item = nit) %>%
  ggplot()+
  geom_boxplot(aes(x=C,y=y.prob.ppc), outlier.colour = NA)+
  geom_point(aes(x=C,y=yobs), color="red")+
  lims(y=c(0, 1))+
  labs(y="Posterior Predictive Category Proportion", x="Item Category")+
  facet_wrap(.~nit, nrow=1)+
  theme_bw()+
  theme(
    panel.grid = element_blank(),
    strip.background = element_rect(fill="white")
  )
p

ggsave(filename = "fig/study4_model1_ppc_y.pdf",plot=p,width = 6, height=3,units="in")
ggsave(filename = "fig/study4_model1_ppc_y.png",plot=p,width = 6, height=3,units="in")
ggsave(filename = "fig/study4_model1_ppc_y.eps",plot=p,width = 6, height=3,units="in")

Manuscript Table and Figures

Table

# print to xtable
print(
  xtable(
    model.fit$BUGSoutput$summary,
    caption = c("study4 Model 1 posterior distribution summary")
    ,align = "lrrrrrrrrr"
  ),
  include.rownames=T,
  booktabs=T
)

% latex table generated in R 4.0.5 by xtable 1.8-4 package
% Thu Jan 20 13:28:36 2022
\begin{table}[ht]
\centering
\begin{tabular}{lrrrrrrrrr}
  \toprule
 & mean & sd & 2.5\% & 25\% & 50\% & 75\% & 97.5\% & Rhat & n.eff \\ 
  \midrule
deviance & 704.72 & 30.59 & 647.94 & 684.71 & 703.64 & 723.98 & 763.98 & 1.00 & 1600.00 \\ 
  lambda[1] & 0.51 & 0.23 & 0.10 & 0.34 & 0.50 & 0.66 & 1.00 & 1.01 & 430.00 \\ 
  lambda[2] & 0.66 & 0.25 & 0.21 & 0.48 & 0.64 & 0.81 & 1.18 & 1.00 & 650.00 \\ 
  lambda[3] & 2.00 & 0.66 & 0.92 & 1.52 & 1.93 & 2.37 & 3.52 & 1.03 & 130.00 \\ 
  lambda[4] & 1.66 & 0.60 & 0.69 & 1.23 & 1.59 & 2.03 & 3.01 & 1.02 & 160.00 \\ 
  lambda[5] & 0.65 & 0.25 & 0.18 & 0.47 & 0.63 & 0.81 & 1.18 & 1.01 & 480.00 \\ 
  lambda[6] & 0.83 & 0.35 & 0.23 & 0.59 & 0.80 & 1.03 & 1.60 & 1.01 & 350.00 \\ 
  lambda[7] & 1.05 & 0.43 & 0.27 & 0.75 & 1.01 & 1.31 & 2.03 & 1.01 & 450.00 \\ 
  lambda[8] & 1.53 & 0.61 & 0.56 & 1.10 & 1.45 & 1.86 & 3.02 & 1.00 & 760.00 \\ 
  lambda[9] & 0.75 & 0.31 & 0.23 & 0.54 & 0.72 & 0.92 & 1.45 & 1.00 & 580.00 \\ 
  lambda[10] & 1.57 & 0.51 & 0.74 & 1.20 & 1.50 & 1.85 & 2.78 & 1.01 & 310.00 \\ 
  lambda.std[1] & 0.43 & 0.16 & 0.10 & 0.32 & 0.44 & 0.55 & 0.71 & 1.01 & 430.00 \\ 
  lambda.std[2] & 0.53 & 0.15 & 0.21 & 0.43 & 0.54 & 0.63 & 0.76 & 1.00 & 700.00 \\ 
  lambda.std[3] & 0.87 & 0.08 & 0.68 & 0.84 & 0.89 & 0.92 & 0.96 & 1.02 & 330.00 \\ 
  lambda.std[4] & 0.82 & 0.10 & 0.57 & 0.77 & 0.85 & 0.90 & 0.95 & 1.01 & 210.00 \\ 
  lambda.std[5] & 0.52 & 0.15 & 0.18 & 0.43 & 0.54 & 0.63 & 0.76 & 1.01 & 590.00 \\ 
  lambda.std[6] & 0.60 & 0.16 & 0.22 & 0.51 & 0.63 & 0.72 & 0.85 & 1.01 & 400.00 \\ 
  lambda.std[7] & 0.68 & 0.16 & 0.26 & 0.60 & 0.71 & 0.79 & 0.90 & 1.01 & 630.00 \\ 
  lambda.std[8] & 0.80 & 0.12 & 0.49 & 0.74 & 0.82 & 0.88 & 0.95 & 1.02 & 540.00 \\ 
  lambda.std[9] & 0.57 & 0.15 & 0.23 & 0.47 & 0.59 & 0.68 & 0.82 & 1.00 & 710.00 \\ 
  lambda.std[10] & 0.82 & 0.09 & 0.60 & 0.77 & 0.83 & 0.88 & 0.94 & 1.01 & 350.00 \\ 
  reli.omega & 0.92 & 0.02 & 0.88 & 0.91 & 0.93 & 0.94 & 0.95 & 1.01 & 370.00 \\ 
  tau[1,1] & -0.96 & 0.18 & -1.33 & -1.07 & -0.95 & -0.83 & -0.62 & 1.00 & 1400.00 \\ 
  tau[2,1] & -0.13 & 0.17 & -0.46 & -0.24 & -0.13 & -0.02 & 0.20 & 1.00 & 910.00 \\ 
  tau[3,1] & -2.50 & 0.58 & -3.90 & -2.82 & -2.43 & -2.08 & -1.61 & 1.03 & 140.00 \\ 
  tau[4,1] & -3.06 & 0.64 & -4.57 & -3.40 & -2.98 & -2.61 & -2.05 & 1.02 & 200.00 \\ 
  tau[5,1] & -0.15 & 0.17 & -0.48 & -0.26 & -0.15 & -0.04 & 0.17 & 1.00 & 4000.00 \\ 
  tau[6,1] & -2.13 & 0.30 & -2.80 & -2.31 & -2.10 & -1.91 & -1.62 & 1.01 & 510.00 \\ 
  tau[7,1] & -2.80 & 0.46 & -3.85 & -3.06 & -2.75 & -2.48 & -2.05 & 1.00 & 1600.00 \\ 
  tau[8,1] & -3.77 & 0.81 & -5.77 & -4.19 & -3.62 & -3.19 & -2.55 & 1.00 & 2200.00 \\ 
  tau[9,1] & -1.48 & 0.23 & -1.97 & -1.62 & -1.47 & -1.33 & -1.08 & 1.01 & 470.00 \\ 
  tau[10,1] & -2.32 & 0.46 & -3.40 & -2.58 & -2.26 & -2.00 & -1.60 & 1.01 & 610.00 \\ 
  theta[1] & 1.31 & 0.28 & 1.01 & 1.12 & 1.25 & 1.43 & 2.01 & 1.00 & 620.00 \\ 
  theta[2] & 1.49 & 0.37 & 1.05 & 1.23 & 1.41 & 1.66 & 2.40 & 1.00 & 650.00 \\ 
  theta[3] & 5.42 & 2.96 & 1.85 & 3.32 & 4.72 & 6.63 & 13.36 & 1.03 & 100.00 \\ 
  theta[4] & 4.10 & 2.27 & 1.48 & 2.50 & 3.52 & 5.13 & 10.06 & 1.02 & 160.00 \\ 
  theta[5] & 1.48 & 0.37 & 1.03 & 1.22 & 1.40 & 1.65 & 2.39 & 1.01 & 340.00 \\ 
  theta[6] & 1.81 & 0.68 & 1.05 & 1.34 & 1.65 & 2.07 & 3.55 & 1.01 & 360.00 \\ 
  theta[7] & 2.28 & 1.04 & 1.07 & 1.57 & 2.03 & 2.70 & 5.11 & 1.01 & 340.00 \\ 
  theta[8] & 3.72 & 2.24 & 1.32 & 2.22 & 3.10 & 4.48 & 10.12 & 1.00 & 1200.00 \\ 
  theta[9] & 1.65 & 0.54 & 1.05 & 1.29 & 1.52 & 1.85 & 3.11 & 1.01 & 450.00 \\ 
  theta[10] & 3.72 & 1.87 & 1.55 & 2.45 & 3.26 & 4.41 & 8.72 & 1.01 & 300.00 \\ 
   \bottomrule
\end{tabular}
\caption{study4 Model 1 posterior distribution summary} 
\end{table}

Figure

sessionInfo()

R version 4.0.5 (2021-03-31)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 22000)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.1252 
[2] LC_CTYPE=English_United States.1252   
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.1252    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] car_3.0-10           carData_3.0-4        mvtnorm_1.1-1       
 [4] LaplacesDemon_16.1.4 runjags_2.2.0-2      lme4_1.1-26         
 [7] Matrix_1.3-2         sirt_3.9-4           R2jags_0.6-1        
[10] rjags_4-12           eRm_1.0-2            diffIRT_1.5         
[13] statmod_1.4.35       xtable_1.8-4         kableExtra_1.3.4    
[16] lavaan_0.6-7         polycor_0.7-10       bayesplot_1.8.0     
[19] ggmcmc_1.5.1.1       coda_0.19-4          data.table_1.14.0   
[22] patchwork_1.1.1      forcats_0.5.1        stringr_1.4.0       
[25] dplyr_1.0.5          purrr_0.3.4          readr_1.4.0         
[28] tidyr_1.1.3          tibble_3.1.0         ggplot2_3.3.5       
[31] tidyverse_1.3.0      workflowr_1.6.2     

loaded via a namespace (and not attached):
 [1] minqa_1.2.4        TAM_3.5-19         colorspace_2.0-0   rio_0.5.26        
 [5] ellipsis_0.3.1     ggridges_0.5.3     rprojroot_2.0.2    fs_1.5.0          
 [9] rstudioapi_0.13    farver_2.1.0       fansi_0.4.2        lubridate_1.7.10  
[13] xml2_1.3.2         codetools_0.2-18   splines_4.0.5      mnormt_2.0.2      
[17] knitr_1.31         jsonlite_1.7.2     nloptr_1.2.2.2     broom_0.7.5       
[21] dbplyr_2.1.0       compiler_4.0.5     httr_1.4.2         backports_1.2.1   
[25] assertthat_0.2.1   cli_2.3.1          later_1.1.0.1      htmltools_0.5.1.1 
[29] tools_4.0.5        gtable_0.3.0       glue_1.4.2         reshape2_1.4.4    
[33] Rcpp_1.0.7         cellranger_1.1.0   jquerylib_0.1.3    vctrs_0.3.6       
[37] svglite_2.0.0      nlme_3.1-152       psych_2.0.12       xfun_0.21         
[41] ps_1.6.0           openxlsx_4.2.3     rvest_1.0.0        lifecycle_1.0.0   
[45] MASS_7.3-53.1      scales_1.1.1       ragg_1.1.1         hms_1.0.0         
[49] promises_1.2.0.1   parallel_4.0.5     RColorBrewer_1.1-2 curl_4.3          
[53] yaml_2.2.1         sass_0.3.1         reshape_0.8.8      stringi_1.5.3     
[57] highr_0.8          zip_2.1.1          boot_1.3-27        rlang_0.4.10      
[61] pkgconfig_2.0.3    systemfonts_1.0.1  evaluate_0.14      lattice_0.20-41   
[65] labeling_0.4.2     tidyselect_1.1.0   GGally_2.1.1       plyr_1.8.6        
[69] magrittr_2.0.1     R6_2.5.0           generics_0.1.0     DBI_1.1.1         
[73] foreign_0.8-81     pillar_1.5.1       haven_2.3.1        withr_2.4.1       
[77] abind_1.4-5        modelr_0.1.8       crayon_1.4.1       utf8_1.1.4        
[81] tmvnsim_1.0-2      rmarkdown_2.7      grid_4.0.5         readxl_1.3.1      
[85] CDM_7.5-15         pbivnorm_0.6.0     git2r_0.28.0       reprex_1.0.0      
[89] digest_0.6.27      webshot_0.5.2      httpuv_1.5.5       textshaping_0.3.1 
[93] stats4_4.0.5       munsell_0.5.0      viridisLite_0.3.0  bslib_0.2.4       
[97] R2WinBUGS_2.1-21

Study 4: Extroversion Data Analysis

Model 1 Results

R. Noah Padgett

2022-01-17